I’ve taught topology many times. One of the highlights for the students (and for me) is the investigation of the Möbius band—the one sided, one edged, non-orientable surface with boundary. On the day we introduce the Möbius band I bring many strips of paper, clear tape, and scissors and have the students make conjectures about what would happen if we taped and cut apart various topological shapes. Here are some activities that are fun to do:
- Twist the paper zero times, and tape the ends (making a cylinder). Cut down the midline.
- Give the paper one half-twist, and tape the ends (making a Möbius band). Cut down the midline.
- Give the paper two half-twists ,and tape the ends. Cut down the midline.
- Give the paper three half-twists, and tape the ends. Cut down the midline.
- Twist the paper zero times, and tape the ends. Cut into thirds.
- Give the paper one half-twist, and tape the ends. Cut into thirds.
- Give the paper two half-twists, and tape the ends. Cut into thirds.
- Give the paper three half-twists, and tape the ends. Cut into thirds.
In fact, these activities are fun for people of any age. My senior math majors enjoy it, and my kids’ kindergarten classes have too.
Last week I attended the 12th biennial Gathering 4 Gardner conference—a wonderful meeting of people interested in mathematics, puzzles, games, magic, and skepticism. One of the speakers (Iwahiro Hirokazu Iwasawa) suggested making zip-apart Möbius bands. Genius! And perfect timing (since I’m teaching topology this semester).
When I got home I bought zippers and Velcro and used them to make reusable strips. Half of them were just one zipper with Velcro on the ends. These can be used to do activities 1-4 above. Then I took identical pairs of zippers and joined them side-by side to make strips for activities 5-8.
- The zippers I used were 12″ or 14″ long. They seemed to work well.
- The Velcro has adhesive on the back, so I could just stick them to the ends of the strips. I did it so that the velcro folded over the ends and so was on both sides. That gave me flexibility if I gave an even or an odd number of half twists.
- For the doubled zippers, align them so that both zippers start at the same end of the strip and so that one is on the top side and one is on the bottom side. Then, when you join with an odd number of half-twists, one begins where the other ends and they’re both on the same “side” (locally) of the Möbius band.
- Ideally I would have sewn the two zippers side-by-side. But we don’t own a sewing machine. So I stapled them.
- I learned the hard way that there are separating zippers (think of the zipper on a parka that comes apart at the bottom) and closed-end zippers (think the zipper on your pants that stops at the bottom). You want the former type for this activity.
- The zippers were about $3 apiece, but my wife told me about a local fabric store that was going out of business, so I got them for 70% off!
Video of me unzipping a Möbius band down its center
Video of me unzipping a Möbius band in thirds